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In this book, based on 16 years of work on the finiteelement method, the author presents the essence of a new,direct approach to the FEM. The work is focused on themixed method and shows how reliable results may be obtainedwith fewer equations than usual.The basic principles, the fundamentals and the essence ofthe FEM are presented, then the method is applied to theanalysis of one, two, and three-dimensional problems. It isshown that mixed elements offer superior accuracy comparedwith stiffness elements. Finally, some new achievementsand perspectives for further development are presented.The book is intended for undergraduate and graduatestudents, mathematicians, research engineers and practicingengineers. To understand the book, a familiarity withclassical mechanics is sufficient.